# Write the Distance Between the Vertex and Focus of the Parabola Y2 + 6y + 2x + 5 = 0. - Mathematics

Write the distance between the vertex and focus of the parabola y2 + 6y + 2x + 5 = 0.

#### Solution

Given:

$y^2 + 6y + 2x + 5 = 0$

$\Rightarrow \left( y + 3 \right)^2 + 2x - 4 = 0$
$\Rightarrow \left( y + 3 \right)^2 = - 2\left( x - 2 \right) \left( 1 \right)$

Let Y = y+3,  $X = x - 2$

From (1), we have:

$Y^2 = - 2X$

Putting $4a = 2$

$a = \frac{1}{2}$

Focus = $\left( X = \frac{- 1}{2}, Y = 0 \right) = \left( x = \frac{3}{2}, y = - 3 \right)$

Vertex = $\left( X = 0, Y = 0 \right) = \left( x = 2, y = - 3 \right)$

Thus, we have:
Focus =$\left( \frac{3}{2}, - 3 \right)$

Vertex = $\left( 2, - 3 \right)$

Distance between the vertex and the focus:

$\sqrt{\left( \frac{3}{2} - 2 \right)^2 + \left( - 3 + 3 \right)^2}$
$\sqrt{\left( \frac{1}{2} \right)^2}$
$= \frac{1}{2} \text{ units }$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 25 Parabola
Exercise 25.2 | Q 2 | Page 28